Analysis of a high-order unfitted finite element method for elliptic interface problems

Abstract

In the context of unfitted finite element discretizations the realization of high order methods is challenging due to the fact that the geometry approximation has to be sufficiently accurate. We consider a new unfitted finite element method which achieves a high order approximation of the geometry for domains which are implicitly described by smooth level set functions. The method is based on a parametric mapping which transforms a piecewise planar interface reconstruction to a high order approximation. Both components, the piecewise planar interface reconstruction and the parametric mapping are easy to implement. In this paper we present an a priori error analysis of the method applied to an interface problem. The analysis reveals optimal order error bounds for the geometry approximation and for the finite element approximation, for arbitrary high order discretization. The theoretical results are confirmed in numerical experiments.

An Unfitted Hybrid High-Order Method for Elliptic Interface Problems

“…so that n T (V, V ) ≥ 1 2 (ξ 2 + ζ 2 ) (i.e., (26)) follows from the assumption that η ≥ 4c 2 dtr . Moreover, using the Cauchy-Schwarz inequality, we infer that…”

“…The so-called cutFEM framework was developed recently in [8] so as to couple different physical models over unfitted interfaces and to discretize PDEs over unfitted embedded submanifolds. The high-order approximation of the geometry of the interface was considered recently in [10] using a boundary correction based on local Taylor expansions and in [26] using an iso-parametric technique, the common objective being to simplify the numerical integration on domains with curved boundaries by allowing a piecewise affine representation of the interface. The cutFEM paradigm has also been applied to a variety of complex flow problems, see, e.g., [27], the recent PhD thesis [30], and references therein.…”

Very high‐order accurate finite volume scheme for the convection‐diffusion equation with general boundary conditions on arbitrary curved boundaries

“…In this regard, developing techniques dedicated to boundary conditions which are prescribed on curved boundaries is of paramount importance to achieve high‐order accurate convergence. The classical technique is based on the isoparametric elements method, but similar techniques have been specifically designed for the finite volume method. For example, the seminal paper of Ollivier‐Gooch and Van Altena introduces a technique to handle smooth curved boundaries based on the constrained least‐squares method for the boundary edges.…”

“…For a short literature review on the topic, the reader is referred to the introduction section in the work of Costa et al, which is summarized in the following. The classical approach to handle with curved boundary conditions is based on the isoparametric element method, which requires, on one side, the introduction of curved mesh elements, and on the other side, nonlinear transformations to map the local curved mesh elements onto the reference polygonal ones. An alternative approach, dedicated to the finite volume method, was initially proposed by Ollivier‐Gooch and Van Altena .…”

Unfitted finite element methods for the heat conduction in composite media with contact resistance

“…The major difficulty in the analysis is to estimate the error coming from the curved interface replaced by a discrete one. Note that, more recently Lehrenfeld and Reusken also considered the geometry error for unfitted methods. In that article, the interface was discretized by the corresponding level set function which is different from that of our method.…”